Problem 29 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=\sqrt{x+1}\\\&g(x)=-\sqrt{x+1}\end{aligned}$$

Step-by-Step Solution

Answer

Sketch the graphs of \(f(x) = \sqrt{x+1}\) and \(g(x) = -\sqrt{x+1}\) on the same axes. First, plot the key points of f(x) at (-1, 0), (0, 1), (3, 2), and (8, 3), resulting in a part of a parabola that opens upwards. Then, plot the key points of g(x) after reflecting f(x) over the x-axis to get (-1, 0), (0, -1), (3, -2), and (8, -3), creating a part of a parabola that opens downwards. Both graphs share the point (-1, 0), and g(x) is a reflection of f(x) across the x-axis.

Step 1: Sketch the graph of f(x) = sqrt(x+1)

To sketch the graph of \(f(x) = \sqrt{x+1}\), we need to identify the domain, range, and any key points. The domain of \(f(x)\) is \(x \geq -1\), since the square root function is only defined for non-negative input values. The range of \(f(x)\) is \(y \geq 0\), as the square root function only gives non-negative output values. Now, we can identify some key points for plotting the graph, such as (-1, 0), (0, 1), (3, 2), and (8, 3). With this information, sketch the graph of \(f(x) = \sqrt{x+1}\), which should be a part of a parabola opening upwards, starting at the point (-1, 0) and increasing.

Step 2: Apply the transformation technique to obtain the graph of g(x) = -sqrt(x+1)

To obtain the graph of \(g(x) = -\sqrt{x+1}\) from the graph of \(f(x) = \sqrt{x+1}\), we must apply a reflection about the x-axis. This transformation flips the graph upside-down (reflects it across the x-axis). Since the domain of both functions is the same, the range of \(g(x)\) will now be \(y \leq 0\). Now, we can apply this transformation to the key points we identified for the graph of \(f(x)\). The transformed points for the graph of \(g(x)\) will be (-1, 0), (0, -1), (3, -2), and (8, -3). With this information, sketch the graph of \(g(x) = -\sqrt{x+1}\), which should be a part of a parabola opening downwards, starting at the point (-1, 0) and decreasing.

Step 3: Combine the graphs of f(x) and g(x)

Now that we have the graphs of both \(f(x) = \sqrt{x+1}\) and \(g(x) = -\sqrt{x+1}\), we can sketch them together on the same axes. The two graphs should be reflections of each other across the x-axis, with the graphs of \(f(x)\) increasing as we move to the right and the graph of \(g(x)\) decreasing as we move to the right. Both graphs share the point (-1, 0) as part of their domain, which should be evident on the combined graph.

Key Concepts

Graphing Functions

Graphing functions involves creating a visual representation of how a function behaves. This is crucial for understanding functions as it provides an intuitive way of looking at changes in their behavior. To graph a function, follow these simple steps:

• Identify the function's domain and range. This sets the stage for where your graph will lie on the x and y axes.
• Find key points. These are points that are crucial for forming the shape of the graph. For example, turning points or intersection points with the axes.
• Plot these points on a coordinate grid. Having a good number of points helps in forming a more complete picture of the graph.
• Connect the points smoothly. For most functions, points should form a smooth curve or line.
• Observe any transformations the function might undergo, such as shifts or reflections, which affect how the graph is plotted.

In the original exercise, we graph two functions: a basic square root function and another affected by a transformation. Seeing them together on the same plane offers insights on how transformations modify function graphs.

Square Root Function

The square root function holds a unique shape due to its definition: it only processes non-negative inputs and outputs. This is due to the fact that square roots of negative numbers are not real, which defines the domain of square root functions.

• Domain: For the function \(f(x) = \sqrt{x+1}\), the domain is \(x \geq -1\), as you only get real results when \(x+1\) is non-negative.
• Range: The range is \(y \geq 0\) since square roots provide non-negative output values.
• Key Points: These include vertices like (-1, 0), where the function starts, and other points like (0, 1), (3, 2), and (8, 3) that help sketch its curvature.
• Graph Shape: The graph of a square root function curls upwards and grows slowly without bounds as \(x\) increases.

Graphing \(f(x) = \sqrt{x+1}\) forms a portion of a parabola opening upward, starting at (or vertexing from) the point (-1, 0), curving gently into the first quadrant.

Reflection Over X-Axis

Reflection about the x-axis involves flipping a graph upside down, mirroring it across the x-axis. This transformation changes the visual representation of a function drastically.

• Transformation: When we reflect \(f(x) = \sqrt{x+1}\) to create \(g(x) = -\sqrt{x+1}\), it involves multiplying the function by -1. Hence, every y-value of \(f(x)\) becomes its negative counterpart, affecting key points.
• Key Point Transformation: Using the original key points of \(f(x)\) such as (-1, 0), they become (-1, 0), (0, -1), (3, -2), and (8, -3) when reflected.
• Graph Shape: The reflection causes the parabola of \(g(x)\) to open downward, starting high and moving downwards as \(x\) increases.
• Range Adjustment: As the function \( -\sqrt{x+1}\) descends, its range alters to \(y \leq 0\), meaning all outputs are non-positive.

This mirror reflection helps understand how transformations impact function graphs beyond raw equations, giving insights into the transformations directly from visualizations.